Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers

Abstract On the unit square, we consider a singularly perturbed convection-diffusion boundary value problem whose solution has two exponential boundary layers. We apply the streamline-diffusion finite element method with piecewise bilinear trial functions on a Shishkin mesh of O(N 2) points and show that the error in the discrete space between the computed solution and the interpolant of the true solution is, uniformly in the diffusion parameter ɛ, of order ɛ 1/2 N –1 lnN + N – 3/2 in the usual streamline-diffusion norm. This includes an L 2-norm error estimate of order O(N – 3/2) in the convection–dominated case ɛ ⩽ N – 1 ln–2 N. As a corollary we prove that the method is convergent of order N –1/2 ln3/2 N (again uniformly in ɛ) in the local L ∞ norm on the fine part of the mesh (i.e., inside the boundary layers). This local L ∞ estimate within the layers can be improved to order ɛ 1/2 N –1/2 ln3/2 N+N –1 ln1/2 N, uniformly in ɛ, away from the corner layer.